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The Mach stem equation and amplification in strongly

nonlinear geometric optics

Jean-François Coulombel, Mark Williams

To cite this version:

Jean-François Coulombel, Mark Williams. The Mach stem equation and amplification in strongly nonlinear geometric optics. American Journal of Mathematics, Johns Hopkins University Press, 2017, 139 (4), pp.967-1046. �hal-01102079�

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The Mach stem equation and amplification

in strongly nonlinear geometric optics

Jean-Fran¸cois Coulombel∗ & Mark Williams† January 12, 2015

Abstract

We study highly oscillating solutions to a class of weakly well-posed hyperbolic initial boundary value problems. Weak well-posedness is associated with an amplification phenomenon of oscillating waves on the boundary. In the previous works [CGW14,CW14], we have rigorously justified a weakly nonlinearregime for semilinear problems. In that case, the forcing term on the boundary has amplitude O(ε2) and oscillates at a frequency O(1/ε). The corresponding exact solution, which has been shown to

exist on a time interval that is independent of ε ∈ (0, 1], has amplitude O(ε). In this paper, we deal with the exact same scaling, namely O(ε2) forcing term on the boundary and O(ε) solution, for quasilinear

problems. In analogy with [CGM03], this corresponds to a strongly nonlinear regime, and our main result proves solvability for the corresponding WKB cascade of equations, which yields existence of approximate solutions on a time interval that is independent of ε ∈ (0, 1]. Existence of exact solutions close to approximate ones is a stability issue which, as shown in [CGM03], highly depends on the hyperbolic system and on the boundary conditions; we do not address that question here.

This work encompasses previous formal expansions in the case of weakly stable shock waves [MR83] and two-dimensional compressible vortex sheets [AM87]. In particular, we prove well-posedness for the leading amplitude equation (the “Mach stem equation”) of [MR83] and generalize its derivation to a large class of hyperbolic boundary value problems and to periodic forcing terms. The latter case is solved under a crucial nonresonant assumption and a small divisor condition.

Contents

1 Introduction 3

1.1 General presentation . . . 3

1.2 The equations and main assumptions . . . 5

1.3 Main result for wavetrains . . . 8

1.4 Main result for pulses . . . 9

CNRS and Universit´e de Nantes, Laboratoire de math´ematiques Jean Leray (UMR CNRS 6629), 2 rue de la Houssini`ere,

BP 92208, 44322 Nantes Cedex 3, France. Email: jean-francois.coulombel@univ-nantes.fr. Research of J.-F. C. was supported by ANR project BoND, ANR-13-BS01-0009-01.

University of North Carolina, Mathematics Department, CB 3250, Phillips Hall, Chapel Hill, NC 27599. USA. Email:

williams@email.unc.edu. Research of M.W. was partially supported by NSF grants number 0701201 and DMS-1001616.

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I Highly oscillating wavetrains 12

2 Construction of approximate solutions: the leading amplitude 12

2.1 Some decompositions and notation . . . 12

2.2 Strictly hyperbolic systems of three equations . . . 13

2.3 Extension to strictly hyperbolic systems of size N . . . 18

2.4 The case with a single incoming phase . . . 19

3 Analysis of the leading amplitude equation 19 3.1 Preliminary reductions . . . 20

3.2 Tame boundedness of the bilinear operator Fper . . . 21

3.3 The iteration scheme . . . 24

3.4 Construction of the leading profile . . . 25

4 Proof of Theorem 1.10 26 4.1 The WKB cascade . . . 26

4.2 Construction of correctors . . . 27

4.3 Proof of Theorem 1.10 . . . 33

4.4 Extension to hyperbolic systems with constant multiplicity . . . 34

II Pulses 37 5 Construction of approximate solutions 37 5.1 Averaging and solution operators . . . 39

5.2 Profile construction and proof of Theorem 1.11 . . . 41

6 Analysis of the amplitude equation 45 6.1 Preliminary reductions . . . 45

6.2 Boundedness of the bilinear operator Fpul . . . 47

6.3 The iteration scheme . . . 50

6.4 Construction of the leading profile . . . 51

6.5 Extension to more general N × N systems . . . 52

III Appendices 55 A Example: the two-dimensional isentropic Euler equations 55 B Formal derivation of the large period limit: from wavetrains to pulses 58 B.1 The large period limit of the amplitude equation (2.19) . . . 58

B.2 What is the correct amplitude equation for Mach stem formation ? . . . 61

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1

Introduction

1.1 General presentation

This article is devoted to the analysis of high frequency solutions to quasilinear hyperbolic initial boundary value problems. Up to now, the rigorous construction of such solutions is known in only a few situations and highly depends on the well-posedness properties of the boundary value problem one considers. In the case where the so-called uniform Kreiss-Lopatinskii condition is satisfied, the existence of exact oscillating solutions on a fixed time interval has been proved by one of the authors in [Wil02], see also [Wil96] for semilinear problems. The asymptotic behavior of exact solutions as the wavelength tends to zero is described in [CGW11] for wavetrains and in [CW13] for pulses. The main difference between the two problems is that in the wavetrains case, resonances can occur between a combination of three phases, giving rise to integro-differential terms in the equation that governs the leading amplitude of the solution1. Resonances do not occur at the leading order2 for pulses, which makes the leading amplitude equation easier to deal with in that case.

In this article, we pursue our study of weakly well-posed problems and consider situations where the uniform Kreiss-Lopatinskii condition breaks down. Let us recall that in that case, high frequency oscillations can be amplified when reflected on the boundary. As far as we know, this phenomenon was first identified by Majda and his collaborators, see for instance [MR83,AM87,MA88] in connection with the formation of specific wave patterns in compressible fluid dynamics. Asymptotic expansions in the spirit of [AM87] are also performed in the recent work [WY14]. In various situations (depending on the scaling of the source terms and on the number of phases), these authors managed to derive an equation that governs the leading amplitude of the solution. Solving the leading amplitude equation in [MR83] and constructing exact and/or approximate oscillating solutions was left open. As far as we know, the rigorous justification of such expansions has not been considered in the literature so far. The present article follows previous works where we have given a rigorous justification of the amplification phenomenon: first for linear problems in [CG10], and then for semilinear problems in [CGW14, CW14]. These previous works considered either linear problems, or a weakly nonlinear regime of oscillating solutions for which the existence and asymptotic behavior of exact oscillating solutions can be studied on a fixed time interval.

The regime considered in [MR83,AM87], and that we shall also consider in this article, goes beyond the one considered in [CGW14,CW14]. In analogy with [CGM03], this regime will be referred to as that of strong oscillations. We extend the analysis of [MR83,AM87] to a general framework, not restricted to the system of gas dynamics, and explain why the problem of vortex sheets considered in [AM87] and the analogous one in [WY14] yield a much simpler equation than the problem of shock waves in [MR83]. We also clarify the causality arguments used in [MR83,AM87] to discard some of the terms in the (formal) asymptotic expansion of the highly oscillating solution. We need however to make a crucial assumption in order to analyze this asymptotic expansion, namely we need to assume that no resonance occurs between the phases. This is no major concern for pulses because interactions are not visible at the leading order, and this may be the reason why this aspect was not mentioned in [MR83]. Resonances can have far worse consequences when dealing with wavetrains, and what saves the day in [AM87, WY14] is that there are too few phases to allow for resonances. This explains why the amplitude equation in [AM87] and the corresponding one in [WY14] reduce to the standard Burgers equation. When the system admits at least

1This is not specific to the boundary conditions and is also true for the Cauchy problem in the whole space, see, e.g.,

[HMR86,JMR95].

2Interactions between pulses associated with different phases need to be considered only when dealing with the construction

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three phases (two incoming and one outgoing), and even in the absence of resonances, the amplification phenomenon gives rise, as in [MR83], to integro-differential terms in the equation for the amplitude that determines the trace of the leading profile. We refer to the latter equation as “the Mach stem equation”, and show how it arises more generally in weakly stable (WR class3) hyperbolic boundary problems with a strongly nonlinear scaling, both in the wavetrain setting, where the equation we derive appears to be completely new, and in the pulse setting, where the equation coincides with the one derived in [MR83].

Our main results establish the well-posedness of the Mach stem equation in both settings, and then use those solvability results to construct approximate highly oscillating solutions on a fixed time interval to the underlying hyperbolic boundary value problems. In the wavetrain case we are able to construct approximate solutions of arbitrarily high order under a crucial nonresonant condition and a small divisor condition; in the pulse setting we construct approximate solutions up to the point at which further “correctors” are too large to be regarded as correctors.

In AppendixBwe compute the formal large period limit of the Mach stem equation for wavetrains, and find a surprising discrepancy (described further below) between that limit and the Mach stem equation for pulses derived in [MR83].

The construction of exact oscillating solutions close to approximate ones is a stability issue that is far from obvious in such a strong scaling. We refer to [CGM03] for indications on possible instability issues and postpone the stability problem in our context to a future work. In any case, it is very likely that no general answer can be given and that stability vs instability of the family of approximate solutions will depend on the system and/or on the boundary conditions, see for instance [CGM04] for further results in this direction.

The precise Mach stem formation mechanism described in [MR83] for reacting shock fronts comes from wave breaking (blow-up) in the ”Mach stem equation”. The numerical simulations in [MR84] suggest that the latter equation displays a similar blow-up phenomenon as the corresponding one for the Burgers equation. Our results show that, in a precise functional setting, the Mach stem equation is a semilinear perturbation of the Burgers equation, which might suggest that the hint in [MR84] is true. However, the semilinear perturbation in the Mach stem equation takes the form of a bilinear Fourier multiplier which makes the rigorous justification of a blow-up result difficult. We also postpone this rigorous justification to a future work.

Notation

Throughout this article, we let Mn,N(K) denote the set of n × N matrices with entries in K = R or C, and we use the notation MN(K) when n = N . We let I denote the identity matrix, without mentioning the dimension. The norm of a (column) vector X ∈ CN is |X| := (X∗X)1/2, where the row vector X∗ denotes the conjugate transpose of X. If X, Y are two vectors in CN, we let X · Y denote the quantity P

jXjYj, which coincides with the usual scalar product in RN when X and Y are real. We often use Einstein’s summation convention in order to make some expressions easier to read.

The letter C always denotes a positive constant that may vary from line to line or within the same line. Dependance of the constant C on various parameters is made precise throughout the text. The sign .means ≤ up to a multiplicative constant.

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1.2 The equations and main assumptions

In the space domain Rd+ := {x = (y, xd) ∈ Rd−1× R : xd > 0}, we consider the quasilinear evolution problem with oscillating source term:

(1.1)          ∂tuε+Pdj=1Aj(uε) ∂juε= 0 , t ≤ T , x ∈ Rd+, b(uε|xd=0) = ε 2G  t, y,ϕ0(t, y) ε  , t ≤ T , y ∈ Rd−1, uε, G|t<0= 0 ,

where the Aj’s belong to MN(R) and depend in a C∞ way on u in a neighborhood of 0 in RN, b is a C∞mapping from a neighborhood of 0 in RN to Rp (the integer p is made precise below), and the source term G is valued in Rp. It is also assumed that b(0) = 0, so that the solution starts from the rest state 0 in negative times and is ignited by the small oscillating source term ε2G on the boundary in positive times. The two main underlying questions of nonlinear geometric optics are:

1. Proving existence of solutions to (1.1) on a fixed time interval (the time T > 0 should be independent of the wavelength ε ∈ (0, 1]).

2. Studying the asymptotic behavior of the sequence uε as ε tends to zero. If we let uappε denote an approximate solution on [0, T′], T≤ T , constructed by the methods of nonlinear geometric optics (that is, solving eikonal equations for phases and suitable transport equations for profiles), how well does uappε approximate uε for ε small ? For example, is it true that

lim

ε→0 kuε− u app

ε kL∞([0,T]×Rd

+)→ 0 ?

The above questions are dealt with in a different way according to the functional setting chosen for the source term G in (1.1). More precisely, we distinguish between:

• Wavetrains, for which G is a function defined on (−∞, T0]×Rd−1×R that is Θ-periodic with respect to its last argument (denoted θ0 later on).

• Pulses, for which G is a function defined on (−∞, T0] × Rd−1× R that has at least polynomial decay at infinity with respect to its last argument.

The answer to the above two questions highly depends on the well-posedness of the linearized system at the origin: (1.2)      ∂tv +Pdj=1Aj(0) ∂jv = f , t ≤ T , x ∈ Rd+, db(0) · v|xd=0 = g , t ≤ T , y ∈ R d−1, v, f, g|t<0 = 0 .

The first main assumption for the linearized problem (1.2) deals with hyperbolicity.

Assumption 1.1 (Hyperbolicity with constant multiplicity). There exist an integer q ≥ 1, some real functions λ1, . . . , λq that are analytic on Rd\ {0} and homogeneous of degree 1, and there exist some positive integers ν1, . . . , νq such that:

∀ ξ = (ξ1, . . . , ξd) ∈ Rd\ {0} , det h τ I + d X j=1 ξjAj(0) i = q Y k=1 τ + λk(ξ) νk .

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Moreover the eigenvalues λ1(ξ), . . . , λq(ξ) are semi-simple (their algebraic multiplicity equals their geo-metric multiplicity) and satisfy λ1(ξ) < · · · < λq(ξ) for all ξ ∈ Rd\ {0}.

For reasons that will be fully explained in Sections2 and4, we make a technical complementary assump-tion.

Assumption 1.2 (Strict hyperbolicity or conservative structure). In Assumption 1.1, either all in-tegers ν1, . . . , νq equal 1 (which means that the operator ∂t + PjAj(0) ∂j is strictly hyperbolic), or A1(u), . . . , Ad(u) are Jacobian matrices of some flux functions f1, . . . , fd that depend in a C∞ way on u in a neighborhood of 0 in RN. In the latter case, Assumption 1.1 holds for all u close to the origin, namely for an open neighborhood O of 0 ∈ RN, there exist an integer q ≥ 1, some real functions λ

1, . . . , λq that are C∞ on O × Rd\ {0} and homogeneous of degree 1 and analytic in ξ, and there exist some positive integers ν1, . . . , νq such that:

dethτ I + d X j=1 ξjAj(u) i = q Y k=1 τ + λk(u, ξ) νk ,

for u ∈ O, τ ∈ R and ξ = (ξ1, . . . , ξd) ∈ Rd\ {0}. Moreover the eigenvalues λ1(u, ξ), . . . , λq(u, ξ) are semi-simple and satisfy λ1(u, ξ) < · · · < λq(u, ξ) for all u ∈ O, ξ = (ξ1, . . . , ξd) ∈ Rd\ {0}.

For simplicity, we restrict our analysis to noncharacteristic boundaries and therefore make the following: Assumption 1.3 (Noncharacteristic boundary). The matrix Ad(0) is invertible and the Jacobian matrix B := db(0) has maximal rank, its rank p being equal to the number of positive eigenvalues of Ad(0) (counted with their multiplicity). Moreover, the integer p satisfies 1 ≤ p ≤ N − 1.

Energy estimates for solutions to (1.2) are based on the normal mode analysis, see, e.g., [BGS07, chapter 4]. We let τ − i γ ∈ C and η ∈ Rd−1 denote the dual variables of t and y in the Laplace and Fourier transform, and we introduce the symbol

A(ζ) := −i Ad(0)−1  (τ − iγ) I + d−1 X j=1 ηjAj(0)   , ζ := (τ − iγ, η) ∈ C × Rd−1. For future use, we also define the following sets of frequencies:

Ξ :=n(τ − iγ, η) ∈ C × Rd−1\ (0, 0) : γ ≥ 0o, Σ :=nζ ∈ Ξ : τ2+ γ2+ |η|2 = 1o, Ξ0 :=

n

(τ, η) ∈ R × Rd−1\ (0, 0)o= Ξ ∩ {γ = 0} , Σ0 := Σ ∩ Ξ0. Two key objects in our analysis are the hyperbolic region and the glancing set that are defined as follows. Definition 1.4. • The hyperbolic region H is the set of all (τ, η) ∈ Ξ0 such that the matrix A(τ, η)

is diagonalizable with purely imaginary eigenvalues.

• Let G denote the set of all (τ, ξ) ∈ R × Rdsuch that ξ 6= 0 and there exists an integer k ∈ {1, . . . , q} satisfying

τ + λk(ξ) = ∂λk

∂ξd

(ξ) = 0 .

If π(G) denotes the projection of G on the first d coordinates (that is, π(τ, ξ) := (τ, ξ1, . . . , ξd−1) for all (τ, ξ)), the glancing set G is G := π(G) ⊂ Ξ0.

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We recall the following result that is due to Kreiss [Kre70] in the strictly hyperbolic case (when all integers νj in Assumption 1.1equal 1) and to M´etivier [M´et00] in our more general framework.

Theorem 1.5 ([Kre70,M´et00]). Let Assumptions 1.1and 1.3 be satisfied. Then for all ζ ∈ Ξ \ Ξ0, the matrix A(ζ) has no purely imaginary eigenvalue and its stable subspace Es(ζ) has dimension p. Further-more, Es defines an analytic vector bundle over Ξ \ Ξ

0 that can be extended as a continuous vector bundle over Ξ.

For all (τ, η) ∈ Ξ0, we let Es(τ, η) denote the continuous extension of Es to the point (τ, η). Away from the glancing set G ⊂ Ξ0, Es(ζ) depends analytically on ζ, see [M´et00]. In particular, it follows from the analysis in [M´et00], see similar arguments in [BGRSZ02,Cou11], that the hyperbolic region H does not contain any glancing point, and Es(ζ) depends analytically on ζ in the neighborhood of any point of H. We now make our weak stability condition precise (recall the notation B := db(0)).

Assumption 1.6 (Weak Kreiss-Lopatinskii condition). • For all ζ ∈ Ξ \ Ξ0, KerB ∩ Es(ζ) = {0}. • The set Υ := {ζ ∈ Σ0 : KerB ∩ Es(ζ) 6= {0}} is nonempty and included in the hyperbolic region H. • There exists a neighborhood V of Υ in Σ, a real valued C∞ function σ defined on V, a basis E1(ζ), . . . , Ep(ζ) of Es(ζ) that is of class C∞ with respect to ζ ∈ V, and a matrix P (ζ) ∈ GLp(C) that is of class C∞ with respect to ζ ∈ V, such that

∀ ζ ∈ V , B E1(ζ) . . . Ep(ζ)= P (ζ) diag γ + i σ(ζ), 1, . . . , 1.

As explained in [CG10,CGW14,CW14], Assumption 1.6 is a more convenient description of the so-called WR class of [BGRSZ02]. Let us recall that this class consists of hyperbolic boundary value problems for which the uniform Kreiss-Lopatinskii condition breaks down ”at first order” in the hyperbolic region4. This class is meaningful for nonlinear problems because it is stable by perturbations of the matrices Aj(0) and of the boundary conditions B.

Our final assumption deals with the phase ϕ0 occuring in (1.1). Assumption 1.7 (Critical phase). The phase ϕ0 in (1.1) is defined by

ϕ0(t, y) := τ t + η · y , with (τ , η) ∈ Υ. In particular, there holds (τ , η) ∈ H.

The shock waves problem considered in [MR83] enters the framework defined by Assumptions 1.1,

1.2, 1.3, 1.6 and 1.7 with the additional difficulty that the space domain has a free boundary. The vortex sheets problem considered in [AM87] and the analogous one in [WY14] violate Assumption 1.3

but these problems share all main features which we consider here. We restrict our analysis to fixed noncharacteristic boundaries mostly for convenience and simplicity of notation.

Our main results deal with the existence of approximate solutions to (1.1). This is the reason why we only make assumptions on the linearized problem at the origin (1.2), and not on the full nonlinear problem (1.1).

4Let us also recall that the uniform Kreiss-Lopatinskii condition is satisfied when KerB ∩ Es(ζ) = {0} for all ζ ∈ Ξ, and

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1.3 Main result for wavetrains

In Part I, we consider the nonlinear problem (1.1) with a source term G that is Θ-periodic with respect to its last argument θ0. As evidenced in several previous works, the asymptotic behavior of the solution uε to (1.1) is described in terms of the characteristic phases whose trace on the boundary equals ϕ0. We thus consider the pairwise distinct roots (and all the roots are real) ω1, . . . , ωM to the dispersion relation

dethτ I + d−1 X j=1 η jAj(0) + ω Ad(0) i = 0 .

To each ωm there corresponds a unique integer km ∈ {1, . . . , q} such that τ + λkm(η, ωm) = 0. We can

then define the following real phases and their associated group velocity:

(1.3) ∀ m = 1, . . . , M , ϕm(t, x) := ϕ(t, y) + ωmxd, vm := ∇λkm(η, ωm) .

We let Φ := (ϕ1, . . . , ϕM) denote the collection of phases. Each group velocity vm is either incoming or outgoing with respect to the space domain Rd+: the last coordinate of vm is nonzero. This property holds because (τ , η) does not belong to the glancing set G.

Definition 1.8. The phase ϕm is incoming if the group velocity vm is incoming (∂ξdλkm(η, ωm) > 0),

and it is outgoing if the group velocity vm is outgoing (∂ξdλkm(η, ωm) < 0).

In all what follows, we let I denote the set of indices m ∈ {1, . . . , M } such that ϕm is incoming, and O denote the set of indices m ∈ {1, . . . , M } such that ϕm is outgoing. Under Assumption1.3, both I and O are nonempty, as follows from [CG10, Lemma 3.1] which we recall later on.

The proof of our main result for wavetrains, that is Theorem1.10below, heavily relies on the nonres-onance assumption below. For later use, we introduce the following notation: if 0 ≤ k ≤ M , we let ZM ;k denote the subset of all α ∈ ZM such that at most k coordinates of α are nonzero. For instance ZM ;1 is the union of the sets Z em, m = 1, . . . , M , where (e1, . . . , eM) denotes the canonical basis of RM. We also introduce the notation:

(1.4) L(τ, ξ) := τ I + d X j=1 ξjAj(0) , L(∂) := ∂t+ d X j=1 Aj(0) ∂j.

The nonresonance assumption reads as follows.

Assumption 1.9 (Nonresonance and small divisor condition). The phases are nonresonant, that is for all α ∈ ZM \ ZM ;1, there holds det L(d(α · Φ)) 6= 0, where α · Φ := α

mϕm.

Furthermore, there exists a constant c > 0 and a real number γ such that for all α ∈ ZM \ ZM ;1 that satisfies αm = 0 for all m ∈ O, there holds

| det L(d(α · Φ))| ≥ c |α|−γ.

Let us note that the small divisor condition is only required for α with nonzero components αm which correspond to incoming phases. If there is only one incoming phase, then there is no such α with at least two nonzero coordinates, and we do not need any small divisor condition. The reason for this simplification will be explained in Sections2 and 3. Our main result reads as follows.

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Theorem 1.10. Let Assumptions 1.1, 1.2, 1.3, 1.6, 1.7, 1.9 be satisfied, let T0 > 0 and consider G ∈ C∞((−∞, T0]t; H+∞(Rd−1y × (R/(Θ Z))θ0)) that vanishes for t < 0. Then there exists 0 < T ≤ T0 and

there exists a unique sequence of profiles (Un)n≥0 in C∞((−∞, T ]; H+∞(Rd+× (R/(Θ Z))M)) that satisfies the WKB cascade (4.1), (4.3) below, and Un|t<0 = 0 for all n ∈ N. In particular, each profile Uk has its θ-spectrum included in the set

ZM I :=



α ∈ ZM/ ∀ m ∈ O , αm = 0 , which means that no outgoing signal is generated at any order.

Furthermore, if for all integers N1, N2 ≥ 0, we define the approximate solution uapp,N1,N2 ε (t, x) := NX1+N2 n=0 ε1+nUn  t, x,Φ(t, x) ε  , then               ∂tuapp,Nε 1,N2+ d X j=1

Aj(uapp,Nε 1,N2) ∂juεapp,N1,N2 = O(εN1+1) , t ≤ T , x ∈ Rd+,

b(uapp,N1,N2 ε |xd=0) = ε 2G  t, y,ϕ0(t, y) ε  + O(εN1+2) , t ≤ T , y ∈ Rd−1, uapp,N1,N2 ε |t<0= 0 ,

where the O(εN1+1) in the interior equation and O(εN1+2) in the boundary conditions are measured

respec-tively in the C((−∞, T ]; HN2(Rd

+))∩L∞((−∞, T ]×Rd+) and C((−∞, T ]; HN2(Rd−1))∩L∞((−∞, T ]×Rd−1) norms.

Of course, the approximate solutions provided by Theorem1.10 become interesting only for N1 ≥ 1, that is, when the remainder O(εN1+2) on the boundary becomes smaller than the source term ε2G.

The spectrum property in Theorem 1.10 is a rigorous justification of the causality arguments used in [AM87, WY14]. Theorem 1.10 will be proved in Part I of this article. In Section 2, we shall derive the so-called leading amplitude (Mach stem) equation from which the leading profile U0 is constructed. Section3is devoted to proving well-posedness for this evolution equation. As far as we know, the bilinear Fourier multiplier that we shall encounter in this equation had not appeared earlier in the geometric optics context and our main task is to prove a tame boundedness estimate for this multiplier. Section 4

is devoted to the construction of the correctors Un, n ≥ 1, and to completing the proof of Theorem 1.10. We refer to AppendixA for a discussion of the two-dimensional isentropic Euler equations, with specific emphasis on Assumption 1.9.

1.4 Main result for pulses

We keep the same notation (1.3) for the phases, but now consider the nonlinear problem (1.1) with a source term G that has ”polynomial decay” with respect to its last argument θ0. This behavior is made precise by introducing the following weighted Sobolev spaces:

Γk(Rd) :=nu ∈ L2(Rd−1y × Rθ) : θα∂y,θβ u ∈ L2(Rd) if α + |β| ≤ k o

. Our second main result reads as follows.

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Theorem 1.11. Let Assumptions 1.1, 1.2, 1.3, 1.6, 1.7 be satisfied. Let k0 denote the smallest integer satisfying k0> (d + 1)/2, and let K0, K1 be two integers such that K0 > 8 + (d + 2)/2, K1− K0 ≥ 2 k0+ 2. Let T0> 0 and consider

G ∈ ∩K0−1

ℓ=0 Cℓ((−∞, T0]t; ΓK1−ℓ(Rdy,θ)) ,

that vanishes for t < 0. Then there exists 0 < T ≤ T0 and there exist profiles U0, U1, U2 vanishing in t < 0 that satisfy the WKB cascade (5.2), (5.4) below. If we define the approximate solution

uappε (t, x) := 2 X n=0 ε1+nUn  t, x,ϕ0(t, y) ε , xd ε  , then               ∂tuappε + d X j=1

Aj(uappε ) ∂juappε = O(ε3) , t ≤ T , x ∈ Rd+, b(uappε |xd=0) = ε 2G  t, y,ϕ0(t, y) ε  + O(ε3) , t ≤ T , y ∈ Rd−1, uappε |t<0= 0 ,

where the O(ε3) in the interior equation and in the boundary conditions are measured respectively in the C((−∞, T ]; L2(Rd

+)) ∩ L∞((−∞, T ] × Rd+) and C((−∞, T ]; L2(Rd−1)) ∩ L∞((−∞, T ] × Rd−1) norms. The exact regularity and decay properties of the profiles are given in (5.9).

The approximate solution provided by Theorem1.11 is constructed, as in [MR83], according to the following line of thought: we expect that the exact solution uε to (1.1) admits an asymptotic expansion of the form uε∼ ε X k≥0 εkUk  t, x,ϕ0(t, y) ε , xd ε  ,

that is either finite up to an order K ≥ 2, or infinite. We plug this ansatz and try to identify each profile Uk. The corrector ε U1 is expected to be negligible with respect to U0, and so on for higher indices. Hence the identification of the profiles is based on some boundedness assumption for the correctors to the leading profile. Of course, such assumptions have to be verified a posteriori when constructing U0, U1 and so on. For instance, Theorem1.10 is based on the assumption that one can decompose uε with profiles in H∞, and we give a rigorous construction of such profiles for which the corrector

ε1+nUn  t, x,Φ(t, x) ε  , is indeed an O(ε1+n) in L((−∞, T ] × Rd +).

In Sections 5 and 6, we give a rigorous construction of the leading profile U0 and of the first two correctors U1, U2 that satisfy all the boundedness and integrability properties on which the derivation of the leading amplitude equation relies. In particular in section 5 we explain why, assuming that the first and second correctors U1, U2 satisfy some boundedness and integrability properties in the stretched variables (t, x, θ0, ξd), the leading profile U0 is necessarily determined by an amplitude equation that is entirely similar to the one in [MR83]. The analysis of Section5 clarifies some of the causality arguments used in [MR83]. This makes the arguments of [MR83] consistent, and one of our achievements is to prove in section6 local well-posedness for the leading amplitude equation derived in [MR83] (which we call the Mach stem equation).

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However, the drawback of this approach is that, surprisingly, it is not consistent with the formal large period limit for wavetrains. More precisely, it seems rather reasonable to expect that the pulse problem is obtained by considering the analogous problem for wavetrains with a period Θ and by letting Θ tend to infinity. In particular, the reader can check that the leading amplitude equations derived in [CW13], resp. [CW14], for quasilinear uniformly stable pulse problems, resp. semilinear weakly stable pulse problems, coincide with the large period limit of the analogous equations obtained in [CGW11], resp. [CGW14], for wavetrains, even though the latter equations include interaction integrals to account for resonances. One could therefore adopt a different point of view and first derive the profile equations for pulses by considering the limit Θ → +∞ for wavetrains, and then study the property of the corresponding approximate solution. Surprisingly, the two approaches do not give the same leading profile U0, as we shall explain in AppendixB. It seems very difficult at this stage to decide which of the two approximate solutions should be the most “physically relevant” since we do not have a nonlinear stability result that would claim that the exact solution uε to (1.1) is close to one of these two approximate solutions on a fixed time interval independent of ε small enough. The clarification of this surprising phenomenon is left to a future work.

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Part I

Highly oscillating wavetrains

2

Construction of approximate solutions: the leading amplitude

2.1 Some decompositions and notation

We recall here some useful results from [CG10] and introduce some notation. Recall that the matrix A(τ , η) is diagonalizable with eigenvalues i ωm, m = 1, . . . , M . The eigenspace of A(τ , η) for i ωmcoincides with the kernel of L(dϕm).

Lemma 2.1 ([CG10]). The (extended) stable subspace Es(τ , η) admits the decomposition

(2.1) Es(τ , η) = ⊕

m∈IKer L(dϕm) ,

and each vector space in the decomposition (2.1) is of real type (that is, it admits a basis of real vectors). Lemma 2.2 ([CG10]). The following decompositions hold

(2.2) CN = ⊕M

m=1Ker L(dϕm) = ⊕Mm=1Ad(0) Ker L(dϕm) , and each vector space in the decompositions (2.2) is of real type.

We let P1, . . . , PM, resp. Q1, . . . , QM, denote the projectors associated with the first, resp. second, decomposition in (2.2). Then for all m = 1, . . . , M , there holds Im L(dϕm) = Ker Qm.

Using Lemma2.2, we may introduce the partial inverse Rm of L(dϕm), which is uniquely determined by the relations

∀ m = 1, . . . , M , RmL(dϕm) = I − Pm, L(dϕm) Rm = I − Qm, PmRm = 0 , RmQm= 0 . When the system is strictly hyperbolic, which is the case considered in Sections 2, 3 and most of Section 4, each vector space Ker L(dϕm) is one-dimensional and M = N . The case of conservative hyperbolic systems with constant multiplicity will be dealt with in Paragraph4.4. In the case of a strictly hyperbolic system, we choose, for all m = 1, . . . , N , a real vector rm that spans Ker L(dϕm). We also choose real row vectors ℓ1, . . . , ℓN, that satisfy

∀ m = 1, . . . , N , ℓmL(dϕm) = 0 ,

together with the normalization ℓmAd(0) rm′ = δmm′. With this choice, the partial inverse Rm and the

projectors Pm, Qm are given by ∀ X ∈ CN, RmX = X m′6=m ℓm′X ωm− ωm′ rm′, PmX = (ℓmAd(0) X) rm, QmX = (ℓmX) Ad(0) rm.

According to Assumption 1.6, KerB ∩ Es(τ , η) is one-dimensional and is therefore spanned by some vector e = Pm∈Iem, em ∈ Span rm (here we have used Lemma 2.1). The vector space B Es(τ , η) is (p − 1)-dimensional and is of real type. We can therefore write it as the kernel of a real linear form (2.3) B Es(τ , η) =X ∈ Cp, b X = 0 ,

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for a suitable nonzero row vector b. Eventually, we can introduce the partial inverse of the restriction of B to the vector space Es(τ , η). More precisely, we choose a supplementary vector space of Span e in Es(τ , η):

(2.4) Es(τ , η) = ˇEs(τ , η) ⊕ Span e .

The matrix B then induces an isomorphism from ˇEs(τ , η) to the hyperplane B Es(τ , η). 2.2 Strictly hyperbolic systems of three equations

For simplicity of notation, we first explain the derivation of the leading amplitude equation in the case of a 3 × 3 strictly hyperbolic system. We keep the notation introduced in the previous paragraph, and we make the following assumption.

Assumption 2.3. The phases ϕ1, ϕ3 are incoming and ϕ2 is outgoing.

Assumption2.3corresponds to the case p = 2 in Assumption1.3(up to reordering the phases). The only other possibility that is compatible with Assumption 1.3 is p = 1, and two phases are outgoing. This case would yield the standard Burgers equation for determining the leading amplitude (see Paragraph2.4

below for a detailed discussion), so we focus on p = 2 which incorporates the main new difficulty.

Let us now derive the WKB cascade for highly oscillating solutions to (1.1). The solution uε to (1.1) is assumed to have an asymptotic expansion of the form

(2.5) uε∼ ε X k≥0 εkUk  t, x,Φ(t, x) ε  ,

where we recall that Φ denotes the collection of phases (ϕ1, ϕ2, ϕ3), and the profiles Uk are assumed to be Θ-periodic with respect to each of their last three arguments θ1, θ2, θ3. Plugging the ansatz (2.5) in (1.1) and identifying powers of ε, we obtain the following first three relations for the Uk’s (see Section 4 for the complete set of relations up to any order):

(a) L(∂θ) U0= 0 ,

(b) L(∂θ) U1+ L(∂) U0+ M(U0, U0) = 0 ,

(c) L(∂θ) U2+ L(∂) U1+ M(U0, U1) + M(U1, U0) + N1(U0, U0) + N2(U0, U0, U0) = 0 , (2.6)

where the differential operators L, M, N1, N2 are defined by: L(∂θ) := L(dϕm) ∂θm, M(v, w) := ∂jϕm(dAj(0) · v) ∂θmw , N1(v, w) := (dAj(0) · v) ∂jw , N2(v, v, w) := 1 2∂jϕm(d 2A j(0) · (v, v)) ∂θmw . (2.7)

The equations (2.6) in the domain (−∞, T ] × Rd+× (R/Θ Z)3 are supplemented with the boundary con-ditions obtained by plugging (2.5) in the boundary conditions of (1.1), which yields (recall B = db(0)):

(a) B U0 = 0 , (b) B U1+ 1 2d 2b(0) · (U 0, U0) = G(t, y, θ0) , (2.8)

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where functions on the left hand side of (2.8) are evaluated at xd= 0 and θ1= θ2 = θ3= θ0. In order to get uε|t<0= 0, as required in (1.1), we also look for profiles Uk that vanish for t < 0.

The derivation of the leading amplitude equation is split in several steps, which we decompose below in order to highlight the (slight) differences in Paragraph 4.4 for the case of systems with constant multiplicity.

Step 1: U0 has mean zero.

According to Assumption 1.9, the phases ϕm are nonresonant. Equation (2.6)(a) thus yields the polarization condition for the leading amplitude U0. More precisely, we expand the amplitude U0 in Fourier series with respect to the θm’s, and (2.6)(a) shows that only the characteristic modes Z3,1 occur in U0. More precisely, we can write

(2.9) U0(t, x, θ1, θ2, θ3) = U0(t, x) + 3 X m=1

σm(t, x, θm) rm,

with unknown scalar functions σm depending on a single periodic variable θm and of mean zero with respect to this variable.

Let us now consider Equation (2.6)(b), and integrate with respect to θ1, θ2, θ3. Using the expression (2.9) of U0, we obtain

(2.10) L(∂) U0 = 0 ,

because the quadratic term M(U0, U0) has zero mean with respect to (θ1, θ2, θ3). Indeed, M(U0, U0) splits as the sum of terms that have one of the following three forms :

⋆)∂θmσmfm(t, x) , ⋆) σm∂θmσm˜rm, ⋆) σm1∂θm2σm2r˜m1m2(m1 6= m2) ,

where ˜rm, ˜rm1m2are constant vectors (whose precise expression is useless), and each of these terms has zero

mean with respect to (θ1, θ2, θ3). Equation (2.10) is supplemented by the boundary condition obtained by integrating (2.8)(a), that is,

(2.11) B U0|xd=0 = 0 .

By the well-posedness result of [Cou05], we get U0 ≡ 0. The goal is now to identify the amplitudes σm’s in (2.9).

Step 2: U0 has no outgoing mode.

We first start by showing σ2 ≡ 0. We first integrate (2.6)(b) with respect to (θ1, θ3) and apply the row vector ℓ2 (which amounts to applying Q2), obtaining

ℓ2L(∂)(σ2r2) + ℓ2  1 Θ2 Z Θ 0 Z Θ 0 M(U0, U0) dθ1dθ3  = 0 .

Since there is no resonance among the phases, integration of the quadratic term M(U0, U0) with respect to (θ1, θ3) only leaves the self-interaction term σ2∂θ2σ2, and the classical Lax lemma [Lax57] for the linear

part5

2L(∂)(σ2r2) gives the scalar equation

∂tσ2+ v2· ∇xσ2+ c2σ2∂θ2σ2 = 0 , c2:=

∂jϕ2ℓ2(dAj(0) · r2) r2 ℓ2r2

.

5In fact, ℓ

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Since v2 is outgoing and σ2 vanishes for t < 0, we obtain σ2 ≡ 0.

The above derivation of the interior equation for σ2 can be performed word for word for the other scalar amplitudes σ1, σ3, because M(U0, U0) also has zero mean with respect to (θ1, θ2) and (θ2, θ3). We thus get

(2.12) ∂tσm+ vm· ∇xσm+ cmσm∂θmσm = 0 , m = 1, 3 , cm :=

∂jϕmℓm(dAj(0) · rm) rm ℓmrm

, but we now need to determine the trace of σm on the boundary {xd= 0}.

Since only σ1, σ3appear in the decomposition (2.9), the leading amplitude U0takes values in the stable subspace Es(τ , η) (Lemma 2.1), and the boundary condition (2.8)(a) yields

σ1(t, y, 0, θ0) r1 = a(t, y, θ0) e1, σ3(t, y, 0, θ0) r3 = a(t, y, θ0) e3,

for a single unknown scalar function a of zero mean with respect to its last argument θ0 (recall that e = e1+ e3 spans the vector space Es(τ , η) ∩ Ker B).

Step 3: U1 has no outgoing mode.

The derivation of the equation that governs the evolution of a comes from analyzing the equations for the first corrector U1. Since (2.6)(c) is more intricate than the corresponding equation in [CGW14], the analysis is starting here to differ from what we did in our previous work [CGW14]. The first corrector U1 reads U1(t, x, θ1, θ2, θ3) = U1(t, x) + 3 X m=1 U1m(t, x, θm) + U1nc(t, x, θ1, θ2, θ3) ,

where U1 represents the mean value with respect to (θ1, θ2, θ3), each U1m incorporates the θm-oscillations and has mean zero, and the spectrum of the ”noncharacteristic” part Unc

1 is a subset of Z3\ Z3,1 due to the nonresonant Assumption1.9. More precisely, U1nc is obtained by expanding (2.6)(b) in Fourier series and retaining only the noncharacteristic modes Z3\ Z3,1. From the expression (2.9) of U

0 (recall U0 ≡ 0 and σ2 ≡ 0), we get

(2.13) L(∂θ) U1nc= −σ1∂θ3σ3∂jϕ3(dAj(0) · r1) r3− σ3∂θ1σ1∂jϕ1(dAj(0) · r3) r1.

In particular, the spectrum of U1ncis a subset of the integers α ∈ Z3 that satisfy α2 = 0 and α1α3 6= 0, so Unc

1 has zero mean when integrated with respect to (θ1, θ3).

Equation (2.6)(b) also shows that the component U12 that carries the θ2-oscillations of U1 satisfies L(dϕ2) ∂θ2U

2

1 = 0, so that U12 can be written as U12 = τ2(t, x, θ2) r2 for an unknown scalar function τ2 of zero mean with respect to θ2.

Let us now consider Equation (2.6)(c). Since U0 only has oscillations in θ1 and θ3, and since there is no resonance among the phases, none of the terms N1(U0, U0), N2(U0, U0, U0) has oscillations in θ2 only. Looking also closely at each term in M(U0, U1), M(U1, U0), we find that both expressions have zero mean with respect to (θ1, θ3), because the only way to generate a θ2-oscillation would be to have a nonzero mode of the form (α1, α2, 0) or (0, α2, α3) with α2 6= 0 in U1nc, but there is no such mode according to (2.13). We thus derive the outgoing transport equation

∂tτ2+ v2· ∇xτ2= 0 , from which we get τ2 ≡ 0.

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Step 4: computation of the nonpolarized components of U11, U13, and compatibility condition. At this stage, we know that the first corrector U1 reads

U1 = U1(t, x, θ1, θ3) = U1(t, x) + U11(t, x, θ1) + U13(t, x, θ3) + U1nc(t, x, θ1, θ3) ,

with U1nc determined by (2.13). Moreover, the nonpolarized part of U11,3 is obtained by considering Equation (2.6)(b) and retaining only the θ1,3 Fourier modes. We get

L(dϕm) ∂θmU

m

1 = −L(∂) (σmrm) − σm∂θmσm∂jϕm(dAj(0) · rm) rm, m = 1, 3 ,

so (I − Pm) U1m, m = 1, 3, is the only zero mean function that satisfies (2.14) (I − Pm) ∂θmU

m

1 = −RmL(∂) (σmrm) − σm∂θmσm∂jϕmRm(dAj(0) · rm) rm, m = 1, 3 .

We now consider the boundary condition (2.8)(b), which we rewrite equivalently as: B U1|xd=0+ B P1U 1 1|xd=0,θ1=θ0 + B P3U 3 1|xd=0,θ3=θ0 +B (I − P1) U11|xd=0,θ1=θ0 + B (I − P3) U 3 1|xd=0,θ3=θ0 + B U nc 1 |xd=0,θ1=θ3=θ0 +1 2(d 2b(0) · (e, e)) a2 = G(t, y, θ 0) .

We differentiate the latter equation with respect to θ0 and apply the row vector b, so that the first line vanishes. We are left with

b B (I − P1) (∂θ1U 1 1)|xd=0,θ1=θ0 + b B (I − P3) (∂θ3U 3 1)|xd=0,θ3=θ0+ b B ∂θ0(U nc 1 |xd=0,θ1=θ3=θ0) +1 2b (d 2b(0) · (e, e)) ∂ θ0(a 2) = b ∂ θ0G .

The first two terms in the first row are computed by using (2.14), and [CG10, Proposition 3.5]. We get (2.15) υ ∂θ0(a

2) − X

Lopa + b B ∂θ0(U

nc

1 |xd=0,θ1=θ3=θ0) = b ∂θ0G ,

where the constant υ and the vector field XLop are defined by

υ := 1 2b (d 2b(0) · (e, e)) −1 2b B R1∂jϕ1(dAj(0) · e1) e1− 1 2b B R3∂jϕ3(dAj(0) · e3) e3, (2.16) XLop := b B (R1e1+ R3e3) ∂t+ d−1 X j=1 b B (R1Aj(0) e1+ R3Aj(0) e3) ∂j = ι (∂τσ(τ , η) ∂t+ ∂ηjσ(τ , η) ∂j) , (2.17)

with ι a nonzero real constant, and σ defined in Assumption1.6. It is also shown in [CG10, Proposition 3.5] that the partial derivative ∂τσ(τ , η) does not vanish, so that, up to a nonzero constant, XLop = ∂t+ w · ∇y for some vector w ∈ Rd−1 (which represents the group velocity associated with the characteristic set of the Lopatinskii determinant).

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The final step in the analysis is to compute the derivative ∂θ0(U

nc

1 |xd=0,θ1=θ3=θ0) arising in (2.15) in

terms of the amplitude a. Restricting (2.13) to the boundary {xd= 0} gives L(∂θ) U1nc|xd=0 = − a(t, y, θ1) (∂θ0a)(t, y, θ3) ∂jϕ3(dAj(0) · e1) e3

− a(t, y, θ3) (∂θ0a)(t, y, θ1) ∂jϕ1(dAj(0) · e3) e1.

Let us expand a in Fourier series with respect to θ0 (recall that a has mean zero): a(t, y, θ0) =

X k∈Z∗

ak(t, y) e2 i π k θ0/Θ.

Then the Fourier series of U1nc reads U1nc(t, x, θ1, θ3) = X k1,k3∈Z∗ uk1,k3(t, x) e 2 i π (k1θ1+k3θ3)/Θ, with uk1,k3(t, y, 0) = −L(k1dϕ1+ k3dϕ3) −1(k 1E1,3+ k3E3,1) , E1,3 := ∂jϕ1(dAj(0) · e3) e1, E3,1:= ∂jϕ3(dAj(0) · e1) e3. (2.18)

Plugging the latter expression in (2.15), we end up with the evolution equation that governs the leading amplitude a on the boundary:

(2.19) υ ∂θ0(a

2) − X

Lopa + ∂θ0Qper[a, a] = b ∂θ0G ,

with

Qper[a,ea] := − X k∈Z     X k1+k3=k, k1 k36=0 b B L(k1dϕ1+ k3dϕ3)−1(k1E1,3+ k3E3,1) ak1eak3     e2 i π k θ0/Θ.

Equation (2.19) is a closed equation for the leading amplitude a on the boundary. It involves the vector field XLopassociated with a characteristic of the Lopatinskii determinant, a Burgers term ∂θ0a

2 and a new quadratic nonlinearity ∂θ0Qper[a, a]. The operator Qpertakes the form of a bilinear Fourier multiplier. Its

above expression may be simplified a little bit by computing the decomposition of L(k1dϕ1+ k3dϕ3)−1 on the basis r1, r2, r3, and by recalling the property b B r1 = b B r3 = 0 (so only the component of L(k1dϕ1+ k3dϕ3)−1 on r2 matters). We obtain:

(2.20) Qper[a,ea] := −b B r2 X k∈Z     X k1+k3=k, k1 k36=0 k1ℓ2E1,3+ k3ℓ2E3,1 k1(ω1− ω2) + k3(ω3− ω2) ak1eak3     e2 i π k θ0/Θ.

Anticipating slightly our discussion in Section3, well-posedness of (2.19) will be linked to arithmetic properties of the phases ϕm, and this is one reason for the small divisor condition in Assumption1.9. This is in sharp contrast with the theory of weakly nonlinear geometric optics for both the Cauchy problem (see [HMR86,JMR93,JMR95] and references therein) and for uniformly stable boundary value problems (see [Wil99, Wil02, CGW11]), where arithmetic properties of the phases do not enter the discussion on the leading profile for the high frequency limit.

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2.3 Extension to strictly hyperbolic systems of size N

The above derivation of Equation (2.19) can be extended without any difficulty to the case of a hyperbolic system of size N provided that Assumption1.9is satisfied. In that case, the number of phases equals N . Steps 1 and 2 of the above analysis extend almost word for word, to the price of changing some notation. Namely, the first relations of the WKB cascade (2.6), (2.8) shows that the leading amplitude U0 reads U0(t, x, θ1, . . . , θN) = U0(t, x) + N X m=1 σm(t, x, θm) rm,

with unknown scalar functions σmdepending on a single periodic variable θmand of mean zero with respect to this variable. The quadratic expression M(U0, U0) still has zero mean with respect to (θ1, . . . , θN) so the nonoscillating part U0 satisfies (2.10) and (2.11), and therefore vanishes. Furthermore, each function σm satisfies the Burgers equation (2.12), which reduces the leading amplitude U0 to

(2.21) U0(t, x, θ1, . . . , θN) = X m∈I

σm(t, x, θm) rm,

where I denotes the set of incoming phases. The boundary condition (2.8)(a) then yields ∀ m ∈ I , σm(t, y, 0, θ0) rm= a(t, y, θ0) em,

for a single unknown scalar function a of zero mean with respect to its last argument θ0 (e =Pm∈Iem spans the vector space Es(τ , η) ∩ Ker B).

Step 3 of the above discussion is unchanged, showing that in the first corrector U1, each profile U1m vanishes when the index m corresponds to an outgoing phase. The noncharacteristic part U1ncis obtained by using the relation

L(∂θ) U1nc= − X m1<m2 m1,m2∈I

σm1∂θm2σm2∂jϕm2(dAj(0) · rm1) rm2 + σm2∂θm1σm1∂jϕm1(dAj(0) · rm2) rm1.

which is the analogue of (2.13).

Step 4 is also unchanged because U1 has no outgoing mode, and when m corresponds to an incom-ing phase, (I − Pm) ∂θmU

m

1 is given by (2.14). Eventually, the boundary condition (2.8)(b) gives the compatibility condition

(2.22) υ ∂θ0(a

2) − X

Lopa + ∂θ0Qper[a, a] = b ∂θ0G ,

with υ := 1 2b (d 2b(0) · (e, e)) −1 2 X m∈I b B Rm∂jϕm(dAj(0) · em) em, (2.23) XLop := X m∈I b B Rmem∂t+ d−1 X j=1 X m∈I b B RmAj(0) em∂j = ι (∂τσ(τ , η) ∂t+ ∂ηjσ(τ , η) ∂j) ,

where ι is a nonzero real constant and the function σ is defined in Assumption 1.6. (Again, [CG10, Proposition 3.5] shows that the partial derivative ∂τσ(τ , η) does not vanish.) The new expression of the

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bilinear Fourier multiplier Qper reads: (2.24) Qper[a,ea] := −

X m∈O b B rm X m1<m2 m1,m2∈I X k∈Z      X km1+km2=k, km1 km2 6=0 km1ℓmEm1,m2 + km2ℓmEm2,m1 km1(ωm1 − ωm) + km2(ωm2− ωm) akm1eakm2      e 2 i π k θ0/Θ, with (2.25) ∀ m1, m2 ∈ I , Em1,m2 := ∂jϕm1(dAj(0) · em2) em1.

The definition (2.24) reduces to (2.20) when N = 3 and Assumption 2.3is satisfied.

2.4 The case with a single incoming phase

In this short paragraph, we explain why the computations in [AM87,WY14] lead to the standard Burgers equation for determining the leading amplitude, and does not incorporate any quadratic nonlinearity of the form (2.20) we have found under Assumption2.3.

The vortex sheets problem considered in [AM87] and the analogous one in [WY14] differ from the framework that we consider here by the fact that the (free) boundary is characteristic. Nevertheless, one can reproduce a similar normal modes analysis for trying to detect violent or neutral instabilities. The two-dimensional supersonic regime considered in [AM87] precludes violent instabilities, but a similar situation to the one encoded in Assumption 1.6occurs6. The situation in [WY14] for three dimensional steady flows is similar, and the corresponding Lopatinskii determinant is computed in [WY13].

The two-dimensional Euler equations form a system of three equations (N = 3), but due to the characteristic boundary (the corresponding matrix Ad(0) has a kernel of dimension 1), the number of phases ϕm on either side of the vortex sheet equals 2. One of them is incoming, and the other is outgoing. In such a situation, there are too few incoming phases to create a nontrivial component Unc

1 for the first corrector U1, so that the bilinear Fourier multiplier Qper vanishes. Though our argument is somehow formal, the reader can follow the computations in [AM87] or in [WY14] and check that they follow the exact same procedure that we have described in our general framework.

3

Analysis of the leading amplitude equation

Our goal in this section is to prove a well-posedness result for the leading amplitude equation (2.19). Up to dividing by nonzero constants, and using the shorter notation θ instead of θ0, the equation takes the form

(3.1) ∂ta + w · ∇ya + c a ∂θa + µ ∂θQper[a, a] = g ,

6The reader will find in [CS04] a detailed analysis of the roots of the associated Lopatinskii determinant, showing that

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where w is a fixed vector in Rd−1, c, µ are real constants, and Qper is a bilinear Fourier multiplier with respect to the periodic variable θ:

(3.2) Qper[a, a] := X k∈Z     X k1+k3=k, k1 k36=0 k1ℓ2E1,3+ k3ℓ2E3,1 k1(ω1− ω2) + k3(ω3− ω2) ak1ak3     e2 i π k θ/Θ.

The source term g in (3.1) belongs to H+∞((−∞, T0]t× Rd−1y × (R/(Θ Z))θ), T0 > 0, and vanishes for t < 0. Furthermore, it has mean zero with respect to the variable θ. Recall that in (3.2), ak denotes the k-th Fourier coefficient of a with respect to θ (which is a function of (t, y)).

Recall that for strictly hyperbolic systems of size N , (2.24) should be substituted for (3.2) in the definition of Qper, while in the particular case p = 1 (one single incoming phase), (3.1) reduces to the standard Burgers equation for which our main well-posedness result, Theorem 3.4 below, is well-known. For simplicity, we thus encompass all cases by studying (3.1), (3.2) and leave to the reader the very minor modifications required for the general case.

3.1 Preliminary reductions

We first introduce the nonzero parameters: δ1 := ω1− ω2 ω3− ω1 , δ3:= ω3− ω2 ω3− ω1 ,

that satisfy δ3= 1 + δ1, and we observe that Qper[a, a] in (3.2) can be written as

Qper[a, a] = −

Θ ℓ2E1,3

2 π (ω3− ω1)Fper(∂θa, a) −

Θ ℓ2E3,1

2 π (ω3− ω1)Fper(a, ∂θa) , where the bilinear operator Fper is defined by:

(3.3) Fper(u, v) :=X k∈Z     X k1+k3=k, k1 k36=0 i uk1vk3 k1δ1+ k3δ3     e2 i π k θ/Θ. The bilinear operator Fper satisfies the following two properties:

(Differentiation) ∂θ(Fper(u, v)) = Fper(∂θu, v) + Fper(u, ∂θv) , (3.4)

(Integration by parts) Fper(u, ∂θv) = − 2 π Θ δ3 u v −δ1 δ3 Fper(∂θu, v) , if u0= v0 = 0 . (3.5)

Using the properties (3.4), (3.5), we can rewrite equation (3.1) as

(3.6) ∂ta + w · ∇ya + c a ∂θa + µ Fper(∂θa, ∂θa) = g ,

with new (harmless) constants c, µ for which we keep the same notation. Our goal is to solve equation (3.6), that is equivalent to (3.1), by a standard fixed point argument. The main ingredient in the proof is to show that the nonlinear term Fper(∂θa, ∂θa) acts as a semilinear term in the scale of Sobolev spaces.

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3.2 Tame boundedness of the bilinear operator Fper

The operator Fper is not symmetric but changing the roles of δ1 and δ3, the roles of the first and second argument of Fper in the estimates below can be exchanged. This will be used at one point in the analysis below. We let Hν := Hν(Rd−1× (R/Θ Z)) denote the standard Sobolev space of index ν ∈ N. The norm is denoted k · kHν. Functions are assumed to take real values. In the proof of Theorem 3.1 below, we

shall also make use of fractional Sobolev spaces on the torus R/Θ Z or on the whole space Rd−1. These are defined by means of the Fourier transform, see, e.g., [BGS07,BCD11]. Our main boundedness result for the operator Fper reads as follows.

Theorem 3.1. There exists an integer ν0 > 1 + d/2 such that, for all ν ≥ ν0, there exists a constant Cν satisfying

(3.7) ∀ u, v ∈ Hν, kFper(∂θu, ∂θv)kHν ≤ Cν kukHν0kvkHν + kukHνkvkHν0.

Estimate (3.7) is tame because the integer ν0is fixed and the right hand side of the inequality depends linearly on the norms kukHν, kvkHν. This will be used in the proof of Theorem3.4below for propagating

the regularity of the initial condition for (3.6) on a fixed time interval.

Proof. We first observe that, provided that Fper(u, v) makes sense, then Fper(u, v) takes real values. This simply follows from observing that

(Fper(u, v))−k = (Fper(u, v))k,

provided that u and v take real values. We are now going to prove a convenient new formulation of Assumption 1.9.

Lemma 3.2. There exist a constant C > 0 and a real number γ0≥ 0 such that ∀ k1, k3∈ Z \ {0} ,

1 |k1δ1+ k3δ3|

≤ C min(|k1|γ0, |k3|γ0) .

Proof of Lemma 3.2. Since in our framework, we have I = {1, 3} and O = {2}, we can apply Assumption

1.9to any (k1, 0, k3) ∈ Z3 with k1k36= 0. We compute

L(d(k1ϕ1+ k3ϕ3)) r2 = (k1(ω1− ω2) + k3(ω3− ω2)) Ad(0) r2,

and the quantity k1(ω1− ω2) + k3(ω3− ω2) cannot vanish for otherwise there would be a nonzero vector in the kernel of L(d(k1ϕ1+ k3ϕ3)). We thus derive the bound

1

|k1(ω1− ω2) + k3(ω3− ω2)|

≤ C kL(d(k1ϕ1+ k3ϕ3))−1k ,

for a suitable constant C that does not depend on k1, k3. The norm of L(d(k1ϕ1+ k3ϕ3))−1 is estimated by combining the lower bound given in Assumption 1.9 for the determinant, and an obvious polynomial bound for the transpose of the comatrix. We have thus shown that there exists a constant C > 0 and a real parameter γ0 (which can be chosen nonnegative without loss of generality), that do not depend on k1, k3, such that

1

|k1(ω1− ω2) + k3(ω3− ω2)|

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Up to changing the constant C, we can rephrase this estimate in terms of the rescaled parameters δ1, δ3:

(3.8) 1

|k1δ1+ k3δ3|

≤ C |(k1, k3)|γ0,

and it only remains to substitute the minimum of |k1|, |k3| for the norm |(k1, k3)| in the right hand side of (3.8).

There are two cases. Either |k1δ1 + k3δ3| > |δ1| > 0, and in that case, it is sufficient to choose C ≥ 1/|δ1| (we use γ0 ≥ 0). Or |k1δ1+ k3δ3| ≤ |δ1|, and we have

|k3| ≤ 1 |δ3||k1δ1+ k3δ3| + 1 |δ3||k1δ1| ≤ 2 |δ1| |δ3||k1| , because k1 is nonzero. Up to choosing a new constant C, (3.8) reduces to

1 |k1δ1+ k3δ3|

≤ C |k1|γ0,

and we can prove the analogous estimate with k3 instead of k1 by the same arguments. This completes the proof of Lemma3.2

The proof of Theorem 3.1 relies on the following straightforward extension of [RR82, Lemma 1.2.2]. The proof of Lemma 3.3is exactly the same as that of [RR82, Lemma 1.2.2], and is therefore omitted. Lemma 3.3. Let K : Rd−1× Z × Rd−1× Z → C be a locally integrable measurable function such that, either sup (ξ,k)∈Rd−1×Z Z Rd−1 X ℓ∈Z |K(ξ, k, η, ℓ)|2dη < +∞ , or sup (η,ℓ)∈Rd−1×Z Z Rd−1 X ℓ∈Z |K(ξ, k, η, ℓ)|2dξ < +∞ . Then the map

(f, g) 7−→ Z Rd−1 X ℓ∈Z K(ξ, k, η, ℓ) f (ξ − η, k − ℓ) g(η, ℓ) dη , is bounded on L2(Rd−1× Z) × L2(Rd−1× Z) with values in L2(Rd−1× Z).

To prove boundedness of the bilinear operator Fper(∂θ·, ∂θ·), we shall apply Lemma 3.3in the Fourier variables. More precisely, for functions u, v in the Schwartz space S(Rd−1× (R/Θ Z)), there holds7:

\ ck(Fper(∂θu, ∂θv))(ξ) = Cst Z Rd−1 X ℓ∈Z,ℓ6∈{0,k} (k − ℓ) ℓ (k − ℓ) δ1+ ℓ δ3 \ ck−ℓ(u)(ξ − η)c[ℓ(v)(η) dη .

Omitting from now on the constant multiplicative factor, we consider the symbol K(k, ℓ) :=

(

(k − ℓ) ℓ/((k − ℓ) δ1+ ℓ δ3) if ℓ 6∈ {0, k} ,

0 otherwise.

7Here we use the notation c

kfor the k-th Fourier coefficient with respect to the variable θ, and the ”hat” notation for the

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We wish to bound the Hν norm: Z Rd−1 X k∈Z h(ξ, k)i2 ν|ck(Fper\(∂θu, ∂θv))(ξ)|2dξ , for ν ∈ N large enough (h·i stands as usual for the Japanese bracket).

Given the parameter γ0 ≥ 0 in Lemma 3.2, we fix an integer ν0 > γ0+ 2 + d/2. We consider two functions χ1, χ2 on Rd× Rdsuch that χ1+ χ2 ≡ 1, and

χ1(ξ, k, η, ℓ) = 0 if h(η, ℓ)i ≥ (2/3) h(ξ, k)i , χ2(ξ, k, η, ℓ) = 0 if h(η, ℓ)i ≤ (1/3) h(ξ, k)i . We first consider the quantity

(3.9) Z Rd−1 X ℓ∈Z χ1(ξ, k, η, ℓ) h(ξ, k)iνK(k, ℓ)c\k−ℓ(u)(ξ − η)c[ℓ(v)(η) dη , which we rewrite as Z Rd−1 X ℓ∈Z χ1(ξ, k, η, ℓ) h(ξ, k)iνK(k, ℓ) h(ξ − η, k − ℓ)iνh(η, ℓ)iν0  h(ξ − η, k − ℓ)iνc\k−ℓ(u)(ξ − η)   h(η, ℓ)iν0c[ ℓ(v)(η)  dη . On the support of χ1, there holds

h(ξ − η, k − ℓ)i ≥ h(ξ, k)i − h(η, ℓ)i ≥ 1

3h(ξ, k)i , and therefore Z Rd−1 X ℓ∈Z χ1(ξ, k, η, ℓ) h(ξ, k)i νK(k, ℓ) h(ξ − η, k − ℓ)iνh(η, ℓ)iν0 2 dη ≤ C Z Rd−1 X ℓ∈Z |K(k, ℓ)|2 h(η, ℓ)i2 ν0 dη .

We now use Lemma3.2to derive the bound (k − ℓ) ℓ (k − ℓ) δ1+ ℓ δ3 = 1 |δ1| ℓ − δ3ℓ2 (k − ℓ) δ1+ ℓ δ3 ≤ C |ℓ|γ0+2, from which we get

sup (ξ,k)∈Rd−1×Z Z Rd−1 X ℓ∈Z χ1(ξ, k, η, ℓ) h(ξ, k)i νK(k, ℓ) h(ξ − η, k − ℓ)iνh(η, ℓ)iν0 2 dη ≤ C Z Rd−1 X ℓ∈Z |ℓ|2 (γ0+2) h(η, ℓ)i2 ν0 dη < +∞ ,

thanks to our choice of ν0. Applying Lemma3.3to the quantity in (3.9), we obtain Z Rd−1 X k∈Z Z Rd−1 X ℓ∈Z χ1(ξ, k, η, ℓ) h(ξ, k)iν K(k, ℓ)c\k−ℓ(u)(ξ − η)c[ℓ(v)(η) dη 2 dξ ≤ C kuk2Hνkvk2Hν0.

Similar arguments yield the bound Z Rd−1 X k∈Z Z Rd−1 X ℓ∈Z χ2(ξ, k, η, ℓ) h(ξ, k)iν K(k, ℓ)c\k−ℓ(u)(ξ − η)c[ℓ(v)(η) dη 2 dξ ≤ C kuk2Hν0kvk2Hν,

and the combination of the two previous inequalities gives the expected estimate Z

Rd−1

X k∈Z

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3.3 The iteration scheme

In view of the boundedness property proved in Theorem3.1, Equation (3.6) is a semilinear perturbation of the Burgers equation (the transport term w · ∇y is harmless), and it is absolutely not surprising that we can solve (3.6) by using the standard energy method with a fixed point iteration. This well-posedness result can be summarized in the following Theorem.

Theorem 3.4. Let ν0 be defined as in Theorem 3.1, and let ν ≥ ν0. Then for all R > 0, there exists a time T > 0 such that for all data a0∈ Hν(Rd−1× (R/Θ Z)) satisfying ka0kHν0 ≤ R, there exists a unique

solution a ∈ C([0, T ]; Hν) to the Cauchy problem: (

∂ta + w · ∇ya + c a ∂θa + µ Fper(∂θa, ∂θa) = 0 , a|t=0= a0.

In particular, if a0∈ H+∞(Rd−1× (R/Θ Z)), then a ∈ C([0, T ]; H+∞) where the time T > 0 only depends on ka0kHν0.

Proof. The proof follows the standard strategy for quasilinear symmetric systems, see for instance [BGS07, chapter 10] or [Tay97, chapter 16], and we solve the Cauchy problem by the iteration scheme

(

∂tan+1+ w · ∇yan+1+ c an∂θan+1+ µ Fper(∂θan, ∂θan) = 0 , an+1|t=0= a0,n+1,

where (a0,n) is a sequence of, say, Schwartz functions that converges towards a0 in Hν, and the scheme is initialized with the choice a0 ≡ a0,0. Given the radius R for the ball in Hν0, we can choose some time T > 0, that only depends on R and ν, such that the sequence (an) is bounded in C([0, T ]; Hν). The uniform bound in C([0, T ]; Hν) is proved by following the exact same ingredients as in the case of the Burgers equation. Contraction in C([0, T ]; L2) is obtained by computing the equation for the difference rn+1:= an+1− an, which reads

(3.10) ∂trn+1+ w · ∇yrn+1+ c an∂θrn+1= −c rn∂θan− µ Fper(∂θrn, ∂θan) − µ Fper(∂θan−1, ∂θrn) . The error terms on the right hand-side are written as

Fper(∂θrn, ∂ θan) = − 2 π Θ δ1 rn∂θan− δ3 δ1 Fper(rn, ∂2 θθan) , Fper(∂θan−1, ∂ θrn) = − 2 π Θ δ3 rn∂θan−1− δ1 δ3 Fper(∂2 θθan−1, rn) , where we have used (3.5).

The final ingredient in the proof is a continuity estimate of the form (3.11) kFper(u, v)kL2 ≤ C min kukHν0−2kvkL2, kukL2 kvkHν0−2

 ,

which we now prove for completeness. We apply the Fubini and Parseval-Bessel Theorems to obtain

kFper(u, v)k2L2 = Θ Z Rd−1 X k∈Z X k1+k3=k, k1 k36=0 1 k1δ1+ k3δ3 uk1vk3 2 dy ,

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and then apply the ℓ1⋆ ℓ2→ ℓ2 continuity estimate to derive kFper(u, v)k2L2 ≤ C Z Rd−1 X k∈Z |k|γ0| |u k| ! X k∈Z |vk|2dy . We then apply the Cauchy-Schwarz inequality and derive the estimate

kFper(u, v)k2L2 ≤ C  sup y∈Rd−1 ku(y, ·)k2Hγ0 +1(R/Θ Z)  kvk2L2,

which yields (3.11) because the integer ν0 in Theorem 3.1 can be chosen larger than (d − 1)/2 + γ0+ 3. (The ”symmetric” estimate is obtained by exchanging the roles of u and v.)

At this stage, we multiply Equation (3.10) by rn+1 and perform integration by parts to derive sup t∈[0,T ] krn+1k2L2 ≤ krn+1|t=0k2L2+ C0T sup t∈[0,T ] krn+1k2L2 + C0T sup t∈[0,T ] krn+1kL2 sup t∈[0,T ] krnkL2,

where the constant C0 is independent of n and follows from the uniform bound for supt∈[0,T ]kankHν.

By classical interpolation arguments, (an) converges towards a weakly in L∞([0, T ]; Hν) and strongly in C([0, T ]; Hν′

), ν′ < ν. Continuity of a with values in Hν is recovered by the standard arguments, see, e.g., [Tay97, Proposition 1.4].

If ν > ν0, it remains to show that the time T only depends on the norm ka0kHν0, and this is where the

tame estimate of Theorem3.1enters the game. More precisely, we follow the same strategy as in [Tay97, Corollary 1.6], and show that the Hν-norm of the solution a satisfies a differential inequality of the form

dka(t)k2 Hν dt ≤ Cν  ka(t)k2Hν0  ka(t)k2Hν,

where Cν is an increasing function of its argument. In particular, boundedness of a(t) in Hν0 on an interval [0, T′), T> 0, implies a unique extension of the solution a ∈ C([0, T); Hν) beyond the time T, which means that the time T of existence for a only depends on ka0kHν0.

3.4 Construction of the leading profile

Theorem3.4 is the cornerstone of the construction of the leading profile U0. Solvability of (2.22) for a is summarized in the following result. Recall that the smoothness assumption for G was made in Theorem

1.10.

Corollary 3.5. There exists T > 0, and a ∈ C∞((−∞, T ]; H+∞(Rd−1× (R/Θ Z))) solution to (2.22) with a|t<0= 0. Furthermore, a has mean value zero with respect to the variable θ.

Proof. Equation (2.22) is easier to solve than the pure Cauchy problem in Theorem 3.4 because we can apply Duhamel’s formula starting from the initial condition a0 = 0. From the assumption of Theorem

1.10, we have G ∈ C∞((−∞, T

0]; H+∞(Rd−1× (R/Θ Z))) with T0 > 0 and G|t<0 = 0, so we can find 0 < T ≤ T0 and a ∈ C((−∞, T ]; H+∞(Rd−1× (R/Θ Z))) solution to (2.22) with a|t<0= 0. Here the time T depends on a fixed norm of G. Then Equation (2.22) yields a ∈ C∞((−∞, T ]; H+∞(Rd−1× (R/Θ Z))) by the standard bootstrap argument and smoothness of G.

For every fixed y, the mean value

a(t, y) := 1 Θ

Z Θ 0

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satisfies the homogeneous transport equation

∂ta + w · ∇ya = 0 , with zero initial condition, and therefore vanishes.

After constructing a on the boundary, we can achieve the construction of the leading profile U0 in the whole domain.

Corollary 3.6. Up to retricting T > 0 in Corollary 3.5, for all m ∈ I, there exists a unique solution σm ∈ C∞((−∞, T ]; H+∞(Rd+× (R/Θ Z))) to (2.12) with σm|t<0= 0 and σm|xd=0 = ema, where the real

number em is defined by em = emrm. Furthermore, each σm has mean value zero with respect to the variable θm.

The result follows from solving the boundary value problem for the Burgers equation (2.12) with prescribed Dirichlet boundary condition on {xd = 0}. This is a (very!) particular case of a quasilinear hyperbolic system with strictly dissipative boundary conditions for which well-posedness follows from the classical theory, see, e.g., [BGS07].

The leading profile U0 is then given by (2.21) and belongs to C∞((−∞, T ]; H+∞(Rd+× (R/Θ Z)N)). Furthermore, its spectrum with respect to the periodic variables (θ1, . . . , θN) is included in the set

(3.12) ZN I :=  α ∈ ZN/ ∀ m ∈ O , αm = 0 , as claimed in Theorem1.10.

4

Proof of Theorem

1.10

4.1 The WKB cascade

In this paragraph, we give a more detailed version of (2.6)-(2.8). We plug again the ansatz (2.5) in (1.1) and derive the set of equations (4.1)-(4.3) below. We recall that the operators L, M are defined in (2.7), while L(∂) is defined in (1.4). Then the WKB cascade in the interior reads:

(a) L(∂θ) U0 = 0 , (b) L(∂θ) U1+ L(∂) U0+ M(U0, U0) = 0 , (c) L(∂θ) Uk+2+ L(∂) Uk+1+ M(U0, Uk+1) + M(Uk+1, U0) + Fk= 0 , k ≥ 0 , (4.1) with ∀ k ≥ 0 , Fk:= ∂jϕm   k+2 X ℓ=2 1 ℓ! X κ1+···+κℓ=k+2−ℓ dℓAj(0) · (Uκ1, . . . , Uκℓ)   ∂θmU0 + k+1 X ℓ=1 Ak+2−ℓ j ∂jUℓ−1+ ∂jϕm k X ℓ=1 Ak+2−ℓ j ∂θmUℓ, (4.2) ∀ ν ≥ 1 , Aν j := ν X ℓ=1 1 ℓ! X κ1+···+κℓ=ν−ℓ dℓAj(0) · (Uκ1, . . . , Uκℓ) ,

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Observe that (4.1)(c) coincides with (2.6)(c) for k = 0. Furthermore, each matrix Aνj, ν ≥ 1, only depends on U0, . . . , Uν−1, and therefore each source term Fk, k ≥ 0, only depends on U0, . . . , Uk.

The set of boundary conditions for (4.1) reads (recall B = db(0)): (a) B U0 = 0 , (b) B U1+ 1 2d 2b(0) · (U 0, U0) = G(t, y, θ0) , (c) B Uk+2+ d2b(0) · (U0, Uk+1) + Gk= 0 , k ≥ 0 , (4.3) with (4.4) ∀ k ≥ 0 , Gk:= k+3 X ℓ=3 1 ℓ! X κ1+···+κℓ=k+3−ℓ dℓb(0) · (Uκ1, . . . , Uκℓ) .

In (4.3) and (4.4), all functions on the left hand side are evaluated at xd = 0, θ1 = · · · = θN = θ0. The source term Gk in (4.3)(c) only depends on U0, . . . , Uk.

We are looking for a sequence of profiles (Uk)k∈N that satisfies (4.1)-(4.3), and Uk|t<0 = 0 for all k.

4.2 Construction of correctors

Some notation will be useful in the arguments below. For any function f that depends on (t, x, θ1, . . . , θN), with Θ-periodicity with respect to each θm, we decompose f as

f = f (t, x) + N X m=1

fm(t, x, θm) + fnc(t, x, θ1, . . . , θN) ,

where f stands for the mean value of f on the torus (R/Θ Z)N, each fm incorporates the θm-modes of f (in particular, the spectrum of fm is included in ZN ;1 and fm has mean zero with respect to θm), and the spectrum of fnc is included in ZN \ ZN ;1. Here, the spectrum only refers to the Fourier decomposition of f with respect to (θ1, . . . , θN). The mappings f 7→ fm and f 7→ fnc are continuous on C∞((−∞, T ]; H+∞(Rd+× (R/Θ Z)N)). Furthermore, if the spectrum of f is included in ZN

I, then fnc belongs to the space of profiles Pnc defined in Lemma4.1below.

The following observation is well-known in the theory of geometric optics, see for instance [JMR93,

Wil99], and relies on Assumption 1.9.

Lemma 4.1. The operator L(∂θ) is a bounded isomorphism from Pnc into itself, where Pnc:=nf ∈ C((−∞, T ]; H+∞(Rd

+× (R/Θ Z)N)) / Spectrum (f ) ⊂ ZNI \ ZN ;1 o

.

Indeed, for α ∈ ZNI \ ZN ;1, the matrix L(d(α · Φ)) is invertible and the norm of its inverse is bounded polynomially in |α| (the degree of the polynomial being fixed). We shall feel free to write L(∂θ)−1fnc when fncis an element of Pnc.

Unsurprisingly, the construction of the sequence (Uk)k∈N is based on an induction process. We formu-late our induction assumption.

(Hn) There exist profiles U0, . . . , Unin C∞((−∞, T ]; H+∞(Rd+× (R/Θ Z)N)) that vanish for t < 0, whose (θ1, . . . , θN)-spectrum is included in ZNI, and that satisfies

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